Abstract: We study large systems of stochastic processes (particles) in which each particle is associated with a vertex in a graph and interacts only with its neighbors. When the graph is complete and the numbers of particles grows to infinity, the system is well-described by a McKean-Vlasov equation, which describes the behavior of one typical particle. For general (sparse) graphs, the system is no longer exchangeable, and the mean field approximation is not valid. Nevertheless, if the underlying graph is locally tree-like, we show that a single particle and its nearest neighbors are characterized by a peculiar but autonomous set of "local dynamics." This work is motivated in part by recent mean field models of inter-bank lending, which capture several dynamic features of systemic risk but thus far lack realistic network structure. Joint work with Kavita Ramanan and Ruoyu Wu.

Bio: Daniel Lacker is an assistant professor in Industrial Engineering and Operations Research (IEOR) at Columbia University. From 2015-2017 he was an NSF postdoctoral fellow in Applied Mathematics at Brown University, and before that he completed his Ph.D. in 2015 at Princeton University in the department of Operations Research and Financial Engineering (ORFE). So far his research has focused largely on the theory and applications of mean field games, where the areas of interacting particle systems, stochastic control, and game theory intersect. More broadly, he is interested in many topics in probability and mathematical finance.